Chains in the Lindenbaum algebra

Question

What is the easiest example of an infinite chain in a Lindenbaum algebra for the propositional calculus?

Does there exist an infinite antichain in a Lindenbaum algebra?

Answer 1

You need to have an infinite supply of variables, because the algebra for classical propositional logic in any finite number of variables is finite.

So say the variables are $\{A, B, C, \ldots \}$. Then there is an:

• Infinite chain: $A \vdash A \lor B \vdash A \lor B \lor C \vdash \cdots $

• Infinite antichain: $\{A, B, C, \ldots\}$.

Apostolos said as much in a comment while I was typing this, so I will make it community wiki.